Yes...but here's a step-by-step for entering them into the website...
Step #1: Take the PHYSICAL cards that you have labeled and enter them into the website, naming them with their corresponding labels.
Step #2: When you get a new card that you want to "load" onto another card, enter it into the website and name it something like "new", or whatever. Then, hit "transfer funds". **BE CAREFUL! If you have more than one card in your account, it will AUTOMATICALLY transfer the funds to your "primary card" unless you select "choose a different card". You can always re-assign a primary card.
Step #3: Once you have transferred the funds from your "new" card, you can delete it from your account. The funds are now secured on the card you transferred them to so there is no reason to keep it.
A few tidbits...
You can only transfer the ENTIRE balance of a gift card. If you have $100 and you want to put $50 into your dining card and $50 into your
DVC card, you MUST have 2 $50 gift cards. For this reason, I actually prefer to accumulate the $25 and $50 cards unless I know I need $100 in a particular account.
I like to make my primary card the card that I will be using soonest. For example, I have a solo trip coming up this October so I have my "Solo Trip" card listed as my primary card. All of my SwagBucks earnings go to this card. I have a family trip planned for next year and that card gets all of my Target FWP cards and BB rewards. After I take my solo trip, then I will move my "Family Trip" card to the primary spot.